Question

Let S be the set of all real numbers of the form

a1/3 + a2/3^2 + a3/3^3 + ....
where a_i \in {0, 1, 2} for all i.

(a) Is the number 1/7 in the set S?

(b) Is the number 1/4 in the set S?

Answer 1

(a) To determine if 1/7 is in the set S, we need to find a sequence of values of a_i such that the sum equals 1/7. We can start by expressing 1/7 as a sum of powers of 3:

1/7 = 3^(-1) + 3^(-2) + 23^(-3) + 23^(-4) + 2*3^(-5) + ...

Since each a_i is either 0, 1, or 2, it is not possible to match this sequence with the sequence defining S. Therefore, 1/7 is not in the set S.

(b) To determine if 1/4 is in the set S, we can again start by expressing 1/4 as a sum of powers of 3:

1/4 = 3^(-1) - 3^(-2) + 23^(-3) + 23^(-4) + 2*3^(-5) + ...

This time, we can match this sequence with the sequence defining S by setting a_1 = 1, a_2 = 2, and a_i = 2 for all i >= 3. This gives:

1/4 = 1/3 - 2/9 + 2/27 + 2/81 + 2/243 + ...

Therefore, 1/4 is in the set S.